On an extremal problem in probability theory

The paper considers the problem of finding the absolute maximum and the absolute minimum of functional (1) where $x$ and $y$ are two dependend vector-valued random variables and $Q(y\mid x)$ is an unknown conditional distribution function. The problem is solved when $I_Q(x)$ is a monotone function of all the variables $x$ and the density functions of the random variables $x$ and $y$ are known.